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If the graphs of two linear functions are perpendicular, describe the relationship between the slopes and the y -intercepts.

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If a horizontal line has the equation f ( x ) = a and a vertical line has the equation x = a , what is the point of intersection? Explain why what you found is the point of intersection.

The point of intersection is ( a ,   a ) . This is because for the horizontal line, all of the y coordinates are a and for the vertical line, all of the x coordinates are a . The point of intersection is on both lines and therefore will have these two characteristics.

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Algebraic

For the following exercises, determine whether the equation of the curve can be written as a linear function.

For the following exercises, determine whether each function is increasing or decreasing.

g ( x ) = 5 x + 6

Increasing

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b ( x ) = 8 3 x

Decreasing

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k ( x ) = −4 x + 1

Decreasing

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p ( x ) = 1 4 x 5

Increasing

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n ( x ) = 1 3 x 2

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m ( x ) = 3 8 x + 3

Decreasing

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For the following exercises, find the slope of the line that passes through the two given points.

( 2 , 4 ) and ( 4 , 10 )

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( 1 , 5 ) and ( 4 , 11 )

2

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( –1 , 4 ) and ( 5 , 2 )

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( 8 , –2 ) and ( 4 , 6 )

–2

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( 6 , 11 ) and ( –4 , 3 )

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For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible.

f ( 5 ) = −4 , and f ( 5 ) = 2

y = 3 5 x 1

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f ( −1 ) = 4 , and f ( 5 ) = 1

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Passes through ( 2 , 4 ) and ( 4 , 10 )

y = 3 x 2

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Passes through ( 1 , 5 ) and ( 4 , 11 )

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Passes through ( −1 , 4 ) and ( 5 , 2 )

y = 1 3 x + 11 3

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Passes through ( −2 , 8 ) and ( 4 , 6 )

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x intercept at ( −2 , 0 ) and y intercept at ( 0 , −3 )

y = 1.5 x 3

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x intercept at ( −5 , 0 ) and y intercept at ( 0 , 4 )

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For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither.

4 x 7 y = 10 7 x + 4 y = 1

perpendicular

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3 y + x = 12 y = 8 x + 1

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3 y + 4 x = 12 6 y = 8 x + 1

parallel

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6 x 9 y = 10 3 x + 2 y = 1

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For the following exercises, find the x - and y- intercepts of each equation.

f ( x ) = x + 2

f ( 0 ) = ( 0 ) + 2 f ( 0 ) = 2 y int : ( 0 , 2 ) 0 = x + 2 x int : ( 2 , 0 )

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h ( x ) = 3 x 5

h ( 0 ) = 3 ( 0 ) 5 h ( 0 ) = 5 y int : ( 0 , 5 ) 0 = 3 x 5 x int : ( 5 3 , 0 )

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2 x + 5 y = 20

2 x + 5 y = 20 2 ( 0 ) + 5 y = 20 5 y = 20 y = 4 y int : ( 0 , 4 ) 2 x + 5 ( 0 ) = 20 x = 10 x int : ( 10 , 0 )

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For the following exercises, use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?

Line 1: Passes through ( 0 , 6 ) and ( 3 , −24 )

Line 2: Passes through ( −1 , 19 ) and ( 8 , −71 )

Line 1: m = –10 Line 2: m = –10 Parallel

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Line 1: Passes through ( −8 , −55 ) and ( 10 , 89 )

Line 2: Passes through ( 9 , 44 ) and ( 4 , 14 )

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Line 1: Passes through ( 2 , 3 ) and ( 4 , −1 )

Line 2: Passes through ( 6 , 3 ) and ( 8 , 5 )

Line 1: m = –2 Line 2: m = 1 Neither

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Line 1: Passes through ( 1 , 7 ) and ( 5 , 5 )

Line 2: Passes through ( −1 , −3 ) and ( 1 , 1 )

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Line 1: Passes through ( 2 , 5 ) and ( 5 , 1 )

Line 2: Passes through ( −3 , 7 ) and ( 3 , −5 )

Line 1 :   m = 2       Line 2 :   m = 2       Parallel

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For the following exercises, write an equation for the line described.

Write an equation for a line parallel to f ( x ) = 5 x 3 and passing through the point ( 2 , 12 ) .

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Write an equation for a line parallel to g ( x ) = 3 x 1 and passing through the point ( 4 , 9 ) .

y = 3 x 3

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Questions & Answers

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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